The rods are a fourth curve

Five of this round’s six departures change the shape of an existing curve. This one adds a curve that was not there, and the difference between those two operations turns out to decide almost everything about how it behaves.

The claim #

A rod contribution is an addition to a three-curve system rather than a modification of it, and additions behave differently from filters in every respect this round measures.

• It cannot be normalised away. A white point divides, and a division does not remove a term that was added; that is why a tenth of a rod signal costs 1.60 ΔE₀₀ where a comparable multiplicative change costs a fraction of that.
• It has the narrowest distribution of the six, spanning a factor of six across forty-two surfaces where the cone optical density spans thirty-five.
• It sits in the wrong place for the cones. Rhodopsin peaks near 500 nanometres, between the short- and middle-wavelength cones, where none of the three is at its maximum.
• And colorimetry has no slot for it. Three numbers is a structural commitment, and a fourth receptor is not a parameter of a three-number system but a violation of it.

What a rod signal actually is #

There are about a hundred and twenty million rods and six million cones, and outside the central degree or two the rods are the large majority. Their photopigment is rhodopsin with a peak near 498 nanometres, and their signal reaches the same bipolar and ganglion cells the cones’ does — there is no separate rod pathway to the brain.

At photopic levels the rods are saturated and contribute nothing. At scotopic levels the cones contribute nothing and there is no colour at all. Between those two, over about three decades of luminance, both are responding and the visual system is receiving four signals through three channels. That is the mesopic range and it is where most evening driving, most street lighting and a good deal of indoor viewing happens.

This collection has been here before from the perception side. Colour goes first in the dark is the phenomenology, and the gamut shrinks in the dark is the consequence for what can be shown. What this round adds is what rod intrusion does to a match — which is a different question, because a match is an identity between numbers rather than a report about appearance.

The model here is the crudest defensible one: a fraction of the scotopic curve summed into all three cone channels, scaled to each channel’s peak. It is a caricature and it is stated as such, and its purpose is to establish the kind of departure rather than its exact size.

Why an addition is not a filter #

The five other departures multiply. A lens absorbs, a macular pigment absorbs, a cone optical density scales the absorbance, and a peak shift moves a curve along the axis — all of them return a modified version of the same three functions.

A rod term does not. It returns lᵢ + k·r where r is a fourth function with its own shape, and no rescaling of lᵢ can produce that. That has three consequences, all of them visible in the measurements.

The white point cannot absorb it. A gain on each cone is invisible because the normalisation divides by the same gain. An addition survives the division: (lᵢ + kr) applied to the sample over (lᵢ + kr) applied to the white does not simplify to the un-added ratio for any k.

Its direction in stimulus space is fixed. A multiplicative change acts differently on samples with different structure; an additive one adds the same vector, scaled only by how much of the rod’s spectral band the sample reflects. That is why its distribution across the surface family is the narrowest of the six.

And it is not a change of observer at all in the strict sense. It is a change of the number of receptors contributing, and a system built on three has no way to express it. The rod term appears in the arithmetic as three modified curves because that is the only shape the arithmetic accepts.

The narrowest distribution, and why that is useful #

Over the forty-two-surface family the rod departure runs from 0.311 to 1.859 ΔE₀₀ with a median of 1.028. Every other departure spans at least a factor of eight and two of them span more than fifteen.

That tightness is worth something practical. A departure with a narrow distribution can be quoted as a single number; one with a wide distribution cannot. A specification budgeting for rod intrusion at a stated luminance can use one figure and be within a factor of two everywhere; a specification budgeting for cone optical density cannot, because the same budget is thirty-five times too large on one sample and correct on another.

The mechanism is worth stating because it generalises. A multiplicative departure’s effect is proportional to how much of the sample’s structure lies where the multiplier changes, which varies enormously between samples. An additive departure’s effect is proportional to how much rod-band light the sample sends, which varies much less — every sample reflects something near 500 nanometres, and few reflect a great deal more than average there. That is the same reason a broad filter has a narrower distribution than a band, arriving through addition rather than through width.

So the shape of a departure’s distribution is a signature of its mechanism, and the two can be told apart from the distribution alone without knowing what either is.

Where the rod curve sits #

Rhodopsin’s peak, seen through the ocular media, is at about 507 nanometres — the CIE’s tabulated V′(λ) peaks there, and this collection’s own construction reproduces it from the pigment template with the macular pigment deliberately left out, because there are no rods in the fovea.

That position is between the short-wavelength cone’s peak at about 440 and the middle-wavelength cone’s at about 541, and it is where the three cone curves are changing fastest relative to one another. So a rod addition is not a small perturbation of any one channel; it is a substantial addition to the middle and short channels and a smaller one to the long, which shifts the balance between them.

The consequence is a systematic blue-green bias in the mesopic range, and it is the Purkinje shift seen from the colorimetric side rather than the photometric one. This collection has measured the photometric version; the colorimetric version is that matches made at photopic levels do not hold at mesopic ones, and the failure is in a fixed direction.

What the standards do about it, which is very little #

Colorimetry’s answer to rod intrusion is to specify that it does not occur: measurements are made at photopic levels, viewing booths are specified at several hundred candelas per square metre, and the standard observers carry no luminance argument at all.

That is a defensible engineering decision and it is a boundary rather than a solution. The CIE has a mesopic photometry system — a two-parameter interpolation between V(λ) and V′(λ) governed by an adaptation level — and it addresses luminance rather than colour. There is no mesopic colorimetry, and the reason is structural: an interpolated luminous efficiency function is still one function, and a fourth receptor needs a fourth coordinate.

A four-receptor system would have a four-dimensional colour space at mesopic levels, and metamerism as everybody knows it — two spectra matching for three numbers — would not apply. Two lights that match at photopic levels would generally not match at mesopic ones, which is exactly what is observed.

That is a much more radical statement than a departure of 1.60 ΔE₀₀ suggests, and the smallness of the number is misleading. The number measures what happens when a fourth signal is forced through a three-channel model; the structural claim is that the model is the wrong shape.

The interaction nobody in this round computed #

There is a coupling between the rod term and the field size that the model carries and the round does not measure, and it is worth naming because it is the largest omission here.

Rod density is zero at the centre of the fovea, rises steeply, and peaks around eighteen degrees out. So a ten-degree field has proportionally far more rod contribution than a two-degree one at the same luminance, and the two parameters are not independent: a field size is two changes at photopic levels and three at mesopic ones.

A large dim field is therefore a substantially different observer from a small dim one, and neither is either standard observer. Anybody specifying a colour for a large surface in low light — a wall, a vehicle interior, a stage — is outside every tabulated observer in a direction none of them is parameterised for.

Computing that interaction needs a rod density profile against eccentricity and a weighting of the field, both of which are available in the literature and neither of which is in this collection’s machinery. It is recorded as owed rather than done.

A fourth curve changes what metamerism is. The structural claim has a consequence for metamerism that is worth drawing out, because it is the clearest way to see that a fourth receptor is not a small correction.

Metamerism is a statement about a null space. Two spectra match when their difference is annihilated by all three sensitivities, and the set of such differences is a subspace of codimension three — enormous, which is why metamers are easy to construct.

Add a fourth sensitivity and the codimension becomes four. The metameric black space shrinks, and every pair that was matching because its difference happened to be orthogonal to three curves now has to be orthogonal to a fourth as well. Almost none of them is.

So a metameric pair constructed for photopic vision generally comes apart at mesopic levels, and it comes apart in the direction the rod curve points. That is a testable prediction and it is the observed behaviour: metameric matches are notoriously condition-dependent, and luminance level is one of the conditions that breaks them, alongside the illuminant and the observer.

What is unusual about this particular breakage is that it cannot be repaired by any three-primary system. A metameric pair that fails because of an illuminant change can be re-matched under the new illuminant; one that fails because of a fourth receptor cannot be re-matched at all with three degrees of freedom, except at one adaptation level.

What was computed, and how #

The rod term is k times the scotopic curve, normalised to each cone’s own peak so that k is a fraction of that channel’s maximum response, summed into all three channels equally. k = 0.10 is the round’s stated strength and it is chosen to correspond roughly to a dim interior rather than fitted to anything.

The scotopic curve is this collection’s own: rhodopsin’s absorbance from the pigment template, through ocular media with the macular pigment removed, normalised to peak at one. It reproduces the CIE’s V′(λ) peak at 507 nanometres from a 498-nanometre pigment, and the eight-nanometre displacement is the lens rather than the molecule.

Everything else is the round’s usual arithmetic, so the number is comparable with the other five and carries the same caveats.

There is a practical setting where all of this is already known and handled badly, and it is worth naming. Museum and archive lighting is specified at low levels to limit photochemical damage — often fifty lux and sometimes less — which is squarely mesopic for a large field. Every colour decision made in such a space is made by a four-receptor system, assessed against a three-receptor standard, and checked with an instrument that has no luminance argument at all.

The usual complaint about low-level viewing is that discrimination falls, which is true and is about thresholds rather than matches. The complaint this round adds is that the match changes, systematically and in a known direction, and that two objects approved as matching at two hundred lux are not necessarily matching at fifty.

Where the model stops #

Equal addition into all three channels is the crudest possible model of how a rod signal reaches the cone pathways, and it is certainly wrong in detail. The physiology has rods feeding cone bipolars through gap junctions with a weighting that is not equal across channels, and the literature reports different rod contributions to the luminance and chromatic pathways.

The model is also linear, and mesopic vision is not. Both rod and cone responses saturate, the balance between them shifts continuously with adaptation, and a fixed fraction is a snapshot of one adaptation state.

And the round measures a match rather than an appearance. Whether two lights differing by a rod contribution look different is a question about the visual system downstream of the receptors, and nothing here bears on it.

That straightness is worth a sentence because it is not obvious. The pairing structure was derived for changes to the sensitivities, and an addition is a change to the sensitivities — lᵢ becomes lᵢ + kr — so the derivation applies unchanged. The departure is still an inner product of the observer’s deviation, which here is kr in every channel, with the stimulus’s deviation from the light.

So the additive departure is structurally the same kind of object as the multiplicative ones and differs only in the direction of the observer’s deviation. That direction is the same in all three channels, which is what makes its effect on the three relative excitations so nearly parallel across samples, and therefore what makes its distribution the tightest of the six.

The generalisation #

The habit is about departures that change the dimension of a model rather than its parameters.

A parameter change keeps a model’s shape and moves a number inside it; a dimension change means the model has no slot for what is happening. Both present as a residual, and a residual can always be reported as a parameter’s worth of error — which is what the 1.60 ΔE₀₀ above is.

The tell is that the residual does not vanish under any setting of the existing parameters. Here that is provable: an addition cannot be produced by any multiplication, so no tuning of the three curves reproduces a rod term.

The failure mode is to fit the residual with the parameters available and report the fit. The model then absorbs a structural error into a parameter that means something else, and the parameter’s value stops being interpretable. A model that fits everything has stopped measuring anything, and the way to avoid it is to ask whether a proposed departure is in the model’s span at all before fitting it.

One consequence for this collection’s own figures is worth recording. Everything drawn here is computed at photopic levels by construction, because no figure family takes a luminance argument except the two that are about adaptation. So none of the collection’s colour results is contaminated by rod intrusion, and none of them applies at mesopic levels either — which is a boundary that has never been stated in a caption.

Who found it, and when #

Purkinje described the shift in relative brightness at twilight in 1825, and the two luminous efficiency functions have been standard since 1924 and 1951 respectively.

Rod intrusion into colour matching was measured through the 1970s and 1980s, notably by Trezona, who published mesopic matching data requiring four primaries — a direct demonstration that the mesopic system is four-dimensional. That work is the reason the structural claim above is not speculation, and it is also why no mesopic colorimetry standard exists: a four-primary matching system does not fit into an apparatus built for three.

Where the ladder goes next #

The remaining departures are filters and shifts, and the two that matter most sit in very specific places on the wavelength axis. A macular pigment is a band rather than a filter, and where a departure lives decides which stimuli it can reach.